Partitions of groups into large subsets

TL;DR

This paper investigates how infinite groups can be partitioned into large subsets based on their algebraic properties, establishing new bounds for such partitions in various types of groups.

Contribution

It introduces new results on partitioning infinite groups into large subsets, including bounds for free groups and general infinite groups based on their cardinality.

Findings

Infinite groups of cardinality k can be partitioned into k left- -large subsets.

Free groups with infinite alphabet k can be partitioned into k 4-large subsets.

Every infinite group can be partitioned into countably many -large subsets.

Abstract

Let G be a group and let k be a cardinal. A subset A of G is called left (right) k-large if there exists a subset F of G such that |F| < { and G = FA (G = AF). We say that A is k-large if A is left and right k-large. It is known that every infinite group G can be partitioned into countably many \aleph_0-large subsets. On the other hand, every amenable (in particular Abelian) group G cannot be partitioned into > \aleph_0 \aleph_0-large subsets. We prove that every infinite group G of cardinality k can be partitioned into k left- \aleph_1-large subsets and every free group F_k in the infinite alphabet k can be partitioned into k 4-large subsets.